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THE AMERICAN SOCIETY OF 
MECHANICAL ENGINEERS 

29 WEST THIRTY-NINTH STREET, NEW YORK 



THE ELASTIC INDENTATION OF STEEL 
BALLS UNDER PRESSURE 

BY 

C. A. BRIGGS 

Mem.Am.Soc.M.E. 

AND 

W. C. CHAPIN and H." G. HEIL 




■ 
- • < 






To bo presented at the Spring Meeting of The American Society of Mechanical 
Engineers, Worcester, Mass., June 4 to 7, 1918. 



The Society as a body is not responsible for the statements of facts or opinions advanced in papers or 

discussions (C 55). 






&> 



THE ELASTIC INDENTATION OF STEEL 
BALLS UNDER PRESSURE 

By C. A. Briggs, Washington, D. C. 

Member of the Society 

and 

W. C. Chapin 1 and H. G. Heil 1 
Non-Members 

This paper reports the results of experiments recently carried out at the Bureau 
of Standards, Washington, D. C, in connection with the adjustment and standard- 
ization of precision apparatus incidental to the testing of munition gages. 

For the case of balls pressing against flat surfaces, which was the one considered 
in the experiments, the results obtained show that the indentation is not directly 
proportional to the pressure but to the two-thirds power of the pressure. This is of 
importance in the design of ball bearings as it indicates what effects are produced 
on the distribution of the load by slight variations in the size of the balls in a 
bearing. 

The results are given in the form of alignment charts and in both metric and 
English units. 

TN the adjustment and standardization of precision apparatus 
at the Bureau of Standards incidental to the testing of muni- 
tion gages, the subject of the effect of pressure on the dimensions of 
steel balls and length standards having rounded ends came up for 
consideration. After a preliminary study of the matter it was con- 
cluded that the distortion of the steel balls between the contacts, and 
the elastic compression of the portion of the rounded ends not directly 
in contact with the measuring surfaces, was very small, a conclusion 
that appeared inconsistent with some of the results that had been 
obtained in actual measurements. This directed attention from the 
main portion of the steel ball or rounded surface to the elastic in- 
dentation of the surfaces immediately in contact with each other. 

1 Division of Weights and Measures, Bureau of Standards, of Washington, 
D.C. 



For presentation at the Spring Meeting, Worcester, Mass., June 1918, of 
The American Society of Mechanical Engineers, 29 West 39th Street, New 
York. All papers are subject to revision. 

3 * 



I 



ELASTIC INDENTATION OF STEEL BALLS UNDER PRESSURE 



In order to settle the questions arising, experiments were undertaken^ 
and the results obtained form the substance of this report. 

2 These experiments, while limited in scope, were so successful 
in giving consistent results and data of apparently wider application 
than that originally intended, that it appears desirable to make a 
brief report summarizing, and recording the information obtained. 





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Fig. 1 Chart for Determining the Indentation of Steel Balls 

(Metric Units) 

Owing to the fact that the effect of high pressure was of no immediate 
interest in reference to the problem in hand, and to the fact that to 
extend the investigation to greater pressures would have required 
time that was needed for other matters, low pressures only were 
employed. 

3 The maximum pressure used on steel balls against glass plates 
was 20 lb., which was used on balls J in. and 1 in. in diameter. On 



TC. A. BRIGGS, W. C. CHAPm AND H. G. HEIL 



smaller balls the maximum pressure used was 10 lb. The results, 
therefore, do not represent conditions present when high pressures 
are used, such as occur in the Brinell hardness test of materials where 
the stresses are above the elastic limit; or pressures such that the 
.indented area includes a solid angle large enough to invalidate the 
assumption frequently made when the angle is :small that the sine 







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Fig. 2 Chart for Determining the Indentation of Steel Balls 

(English Units) 

and angle are equal. However, the results obtained do give accurate 
information of the effect of pressures on the measured diameters of 
spherical surfaces of steel or other materials which are employed in 
standardizing gages ; and also appear to give information of practical 
value as to what occurs in ball bearings. 

4 The experiment was performed by observing and measuring 
with a micrometer microscope the area of contact made by flat and 



6 



ELASTIC INDENTATION OF STEEL BALLS UNDER PRESSURE 



spherical surfaces in contact with each other under varying pressures. 
This area of contact was viewed as the central spot in the Newton's 
ring system formed when a glass surface was in contact with a polished 
surface of steel or with another glass surface. The amount of in- 
dentation was obtained from the measured diameters by a simple 
computation, easily derived, which it is not necessary to give here. 





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Fig. 3 Chart for Determining the Diameter of the Indented Spot 
of Steel Balls (Metric Units) 

5 The different combinations of surfaces available for observa- 
tion and use in the experiments were : a steel sphere pressing against 
a flat glass surface; a spherical glass surface pressing against a flat 
steel surface; and a spherical glass surface pressing against a flat 
glass surface. 

6 From an examination of the results for each pair of surfaces 
in contact plotted on log paper, and from general reasoning based on 



C. A. BRIGGS, W. C. CHAPIN AND H. G. HEIL 7 

the nature of the phenomena, a general equation was worked out for 
the purpose of correlating all of the results. It is not necessary for 
present purposes to expand this report by giving all of the various 
considerations which led to the particular form assumed for this 
general equation — suffice it to state that after the constants of the 
general equation had been determined from the experimental data, 



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Fig. 4 Chart for Determining the Diameter of the Indented Spot of 

Steel Balls (English Units) 



the experimental values were reobtained by computations from the 
general equation, and the agreement between the computed and 
experimental values was very good; in fact, the agreement was so 
satisfactory that it was considered unnecessary to take additional 
data from the same material, as had been contemplated up to that 
time. 



s 



ELASTIC INDENTATION OF STEEL BALLS UNDER PRESSURE 



7 In the case of steel against steel it was of course not possible 
to observe the area of contact owing to the opaque nature of both 
contact surfaces, so that the values of steel against steel were obtained 
by the use of the general formula which had been found to fit all the 
various surfaces and materials used in the experiments. 

8 It will be noted in the equations which follow that a quantity 
called the " indentation modulus" is used to express the elastic 



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Fig. 5 Chart for Determining the Average Pressure per Unit Area on 
the Indented Spot of Steel Balls (Metric Units) 

property which determines the indentation. It was originally in- 
tended to use Young's modulus as representing in a general way the 
elastic properties. However, on looking up the elastic constants of 
glass it was found that a problem of very similar nature had been 
worked out by Hertz from theoretical considerations which gave 
practically the same form of equation as was derived in the present 
experiments, and in which the elastic property effective in determin- 






C. A. BRIGGS, W. C. CHAPIN AND H. G. HEIL 



9 



ing the indentation was found by Hertz to be a function of Young's 
modulus and Poisson's ratio, and which was called the indentation 
modulus. 

9 It has not been necessary for the present purposes to concern 
ourselves with the manner in which the indentation is distributed 






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1,250,000 

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900,000 

300,000 

700,000 
000,000 

500,000 
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Fig. 6 Chart for Determining the Average Pressure per Unit Area on 
the Indented Spot of Steel Balls (English Units) 



between the surfaces in contact, so no further reference will be made 
to this subject. 

10 A very interesting feature which can be noted on examining 
the results is that the indentation is not linear with the pressures but 
is proportional to the two-thirds power of the pressures. This is a 
fact of important interest in connection with the design of ball bear- 



10 ELASTIC INDENTATION OF STEEL BALLS UNDER PRESSURE 

ings, as it indicates what effects are produced on the distribution of 
the load by slight variations in the size of the balls in a bearing. 

11 For steel against steel, which is the case of most general and 
important interest, the results are given in the form of computation 
charts, Figs. 1 to 6, by means of which numerical values can be 
obtained quickly for any particular case covered by the equations. 
Three of the charts are for metric units and three for English units. 

12 In reference to future experiments, those which would appear 
to be of most practical interest are concerned with the determination 
of the elastic indentation of spherical surfaces in contact with cylin- 
drical surfaces and cylindrical surfaces against cylindrical surfaces 
— such as are represented by many forms of ball and roller bearings 
in every-day use. It would be also of interest to extend the experi- 
ments already made by employing higher pressures. 

13 The elastic coefficients for the materials used in the experi- 
ments of this report were obtained from tabular data. If, in the 
future, it becomes desirable to obtain information on the indentation 
of surfaces with great precision, experiments should be performed on 
materials that have had their properties carefully measured, so as to 
fix with exactitude the values of Young's modulus and Poisson's 
ratio for the particular steel and glass used in the experiment. 

14 The results of the experiments are summarized in the eight 
equations which follow. 

15 General Equation for Indentation: 

G\ -f- G^ 



dG = 0.518 P 



E\' ^ EV 



[1] 



where dG = mutual indentation between the surfaces in contact 

P = pressure acting between the two surfaces 
Gi and G 2 = radii of curvature of the two surfaces 
Ei and E 2 f = indentation moduli of the surfaces Gi and 6r 2 . 

The indentation modulus is given by the expression 

E ' = x^ M 

where E = Young's modulus and u = Poisson's ratio. The constant 
0.518 is the same for both metric and English units when either are 
used consistently throughout the formula. 

16 Indentation of Steel Balls Between Flat Steel Surfaces. For 
metric units the indentation is given by the equation 

p! 

2dG = 0.00210^- [3] 

G* 



C. A. BRIGGS, W. C. CHAPIN AND H. G. HEIL 11 

where 2 dG is the indentation in millimeters from both sides of the 
ball. For English units the equation is 

Pi 

2dG = 0.0000166 — x [4] 

G* L J 

where 2 dG is given in inches. 

17 Diameter of Area of Contact Between the Surfaces, Steel Against 
Steel. For metric units, 

R = 0.0647 P*G* [5] 

where R is the diameter of the spot in millimeters. For English 
units, 

R = 0.00576 P$G$ [6] 

where R is given in inches. If the general equation is desired for R 
it can be derived easily from Equation [1]. 

18 Average Pressure Over the Area of Contact, Steel Against Steel. 
For metric units, 

S = 304^ [7] 

G s 

where S is given in kilograms per square millimeter. For English 
units, 

S = 38,400^ [8] 

G s 

where S is given in pounds per square inch. If the general equation 
for S is desired it can likewise be developed from Equation [lj. 

19 Computation Charts. Means for graphically solving the 
preceding equations are shown in the accompanying charts, Figs. 1 
to 6. A straight line placed across any of the charts will strike 
readings on the vertical scales which are a solution of the correspond- 
ing equation. With these charts, when any two of the three quan- 
tities of the equation are given, these quantities will establish two 
points which determine a straight line, and the value of the third 
quantity will be given by the intersection of the straight line with 
the corresponding vertical scale. 




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